The term "interest" is used to indicate
the rent paid for the use of money. It is also used to represent
the percentage earned by an investment in a productive operation.
From the lender's point of view, the interest rate is the ratio
between profit received and investment over a period of time,
which is a contribution to the risk of loss, administrative costs
and pure earnings or profit. From the borrower's point of view,
the interest rate can be expressed as the ratio between the
amount paid for use of the funds and quantity of funds requested.
In this case, the interest to be paid must be less than the
earnings expected.

As money can produce earnings at a certain rate
of interest by being invested for a period of time, it is
important to know that one unit of money received at some future
date does not produce as much earnings as a unit of money
received in the present. This relationship between interest and
time gives rise to the concept of the "time value of money".

Money also has a time value, as its buying power
of a dollar varies with time. During periods of inflation, the
quantity of goods that can be bought with a certain amount of
money decreases as the purchase date moves into the future.
Although this change in the buying power of money is important,
the concept of the time value of money is even more so, in that
it has earning power. Any future reference to the time value of
money will be restricted to this concept. Effects of inflation on
the profitability of an investment are discussed in section 7.9.
It is necessary to know the different methods for computing
interest in order to calculate, with certainty, the actual effect
of the time value of money in the comparison of alternative
courses of action.

Normally, a rate of interest on a sum of money is
expressed as the percentage of the sum that is paid for the use
of the money during a one-year period, but it can also be quoted
for different periods of time. In order to simplify the following
discussion, examination of interest rates for periods other than
one year will be made at the end of this Appendix (13.4). The
interest to be paid on a loan, at simple interest, is
proportional to the principal sum. With P as the principal sum, n
the number of years and i the interest rate, simple interest can
be expressed as:

I = P × n × i
(B.1)

A loan at simple interest can be made for any
period of time. The interest and the initial sum will be paid at
the end of the loan period.

Example B.1 Simple Interest

Find the simple interest on US$ 4 500 at 8% per
year for a) 1 year and b) 4 years.

Answer:

I = P × n × i
I = US$ 4 500 × 1 × 0.08 = US$ 360

The initial sum plus interest increases to US$ 4
860 and will be the total debt at the end of the year.

I = P ×x n × i
I = US$ 4 500 × 4 × 0.08 = US$ 1 440

When calculating the interest owed for a part of
the year, it is usual to consider the year as made up of 12
months of 30 days each, that is, 360 days.

Example B.2 Simple Interest. Period
Length Less than One Year

Find the simple interest on US$ 1000 for the
period 1 February-20 April at 8 % per year.

When interest earned during each period is added
to the principal sum, as shown in Table B.1 (see Example B.3), it
is said that the interest is compounded annually. The difference
between simple interest and compound interest is due to the
effect of capitalization. The final total amount will be higher
when large sums of money, higher rates of interest or a greater
number of periods are involved. Compound interest is the type of
interest that is used in practice, and thus it will be used in
this manual.

Table B.1 Calculation of
Compound Interest

Year

Sum
owed atbeginning of year(A)

Interest
added todebt at year end(B)

Sum
owed atyear end(A + B)

Sum to
be paidat year end

1

1000

1 000 x 0.08 = 80

1 080

0.00

2

1080

1080 x 0.08 = 86.4

1 166.4

0.00

3

1 166.4 1

166.4 x 0.08 = 93.3

1 259.7

0.00

4

1 259.7

1 259.7 x 0.08 = 100.
8

1 360.5

1 360.5

To allow the visualization of each economic
investment alternative a cash expenditure and receipts diagram (Figure
B. 1) is used. This graphic method provides all the information
required to analyse an investment proposal. This diagram shows
income received during the period with an upward arrow (increment
of money) positioned at the end of the period (it is always
assumed to take place at the end of the period in question). The
length of the arrow is proportional to the size of the income
received during the period. Similarly, expenditure is represented
by a downward arrow (a reduction of money). These arrows are
located on a time-scale that represents the duration of the
alternative.

Figures B. 1 (a) and (b) also show the flow
diagrams for the borrower and the lender of the previous example,
where the expenses incurred in initiating an alternative are
deemed to have taken place at the beginning of the period over
which it extends.

In order to evaluate investment alternatives,
sums of money produced at different points in time must be
compared. This is only possible if their characteristics are
analysed on an equivalent basis. Two situations are equivalent
when they have the same effect, the same worth or the same value.
Three factors are involved in the equivalence of investment
alternatives:

the amount of money

the time of occurrence

the rate of interest

The factors of interest which will be developed
take account of duration and rate of interest. Later, they are
used in the transformation of alternatives in terms of a common
time-base.

Interest factors applicable to routine
situations, such as compound interest with single payment, and
with a series of equal payments, will be derived. The following
five points must be kept in mind for application in calculations
of investment alternatives:

The end of one period is, at the same
time, the beginning of the next period.

P is produced at the beginning of a
period, at a time in the present;

F occurs at the end of the nth
period, from a time considered as present (n being the
total number of periods).

A is a single payment within a series of
equal payments made at the end of each period under
consideration. When P and A occur together, the first A
in the series is produced one period after P. When F and
A occur together, the last A in the series occurs at the
same time as F. If the equal payments series occurs at
the beginning of each period under consideration, it is
called Ab.

In proposing different alternatives, the
quantities P, F, A and Ab must be used such
that they incorporate the conditions needed to adjust the
respective models to the factors used.

Table B.2 summarizes the
financial equations that show the relationships between P, F and
A (Jelen and Black, 1983).

If a sum P is invested at the rate of interest i,
how much money is accumulated between capital and interest at the
end of period n; or, what is the equivalent value at the end of
the final period n, of the sum P invested at the beginning of the
operation? The cash flow diagram for this financial situation is
shown in Figure B. 2. Table B. 1 shows interest earned on
applying compound interest to the investment described in Figure
B.2. This investment provides no income during the intermediary
periods. In Table B. 1, the interest earned is added to the
initial sum at the end of each interest period (annual
capitalization). Table B.3 shows the deduction in general terms.

Table B.3 Deduction of Compound-interest
Factor with Single Payment

Year

Amount at beginningof year

Interest earnedduring
year

Compound Interest at
year end

1

P

P × i

P + P × i = P × (1 + i)

2

P × (1+ i)

P × (1 + i) × i

P × (1 + i) + P × (1 +
i) × i = P × (1 + i)2

3

P × (1 + i)2

P × (1 + i)2
× i

P × (1 + i)2
+ P × (1 + i)2 × i = P × (1 + i)3

n

P × (1 + i)n-1

P × (1 + i)n-1
× i

P × (1 + i)n-1
+ P × (1 + i)n-1 × i = P × (1 +i)n
= F

Figure B.2 Single Present Amount
and Single Future Amount

The resulting factor (1 + i)n is known
as the compound-interest factor with single payment, and is
written FPF; the relationship is:

F = P × (1 + i)n
(B. 2)

F = P × FPF
(B. 3)

Example B.4 Single-payment Compound-amount
Factor

Find the compound amount of US$ 1 000 in 4 years
at 8% interest compounded annually.

Answer: From Equation B.3,

F = 1 000 × (1 + 0.08)4 = 1 000
× 1.3605 = US$ 1 360.5

Another way of interpreting Equation B.3 is that
the amount F at some future time is equivalent to the known value
of P at the present time, for the given interest rate of i. The
amount F, US$ 1 360.5, is equivalent to the initial amount P, US$
1 000 at the end of four years if the interest rate is 8 % per
year.

The resulting factor (1 + i)-n is
known as the present-worth factor with single payment and is
written as FFP:

P = F × FFP
(B.5)

Example B.5 Single-payment Present-worth
Factor

How much should be invested now (at present time)
at 8% compound interest per year, in order to receive US$ 1360.5
within 4 years; or what is the present equivalent worth of US$ 1
360.5 to be received four years in the future?

Answer: From Equation B.5,

P = 1 360.5 × (1/1.3605) = 1 360.5 × 0.73503
= US$ 1 000

It is noted that the two factors are reciprocal.
In the Net Present Value and Internal Rate of Return methods used
to evaluate the profitability of projects (Chapter 7), the
present worth factor is applied to compare cash flows with
initial investment.

As an introduction, a definition will be given of
the concept of annuity, which consists of a series of equal
payments made at regular intervals of time, whether annually or
at different periods. This scheme arises in situations such as
accumulation of a determined capital (receipt of a certain lump
sum after a certain amount of periodic payments, as occurs in
some life insurance plans), or cancellation of a debt. Figure B.3
shows the first situation, given that the future value is being
sought, through a series of equal payments made at the end of
successive interest periods.

Figure B.3 Single Future Amount
with Equal-payment Series

The sum of the compound interest of the different
payments can be calculated through the use of the compound-interest
factor with equal payment series. The method of calculating the
factor is to use the compound-interest with single payment factor
to transform each A into its future value:

Amount P is deposited at a present time, at an
interest rate of i per year. The depositor wishes to withdraw the
capital plus the interest earned in a series of equal year end
receipts over the next n years. When the last withdrawal is made,
no funds remain in the deposit. Moreover, it can be said that
whatever the uniform payment is at the end of each period, it is
equivalent to the amount invested at the beginning of the first
year. The cash flow diagram is shown in Figure B.4. To calculate
this factor, it should be expressed as the product of two factors
which are already known, compound-interest factor with single-payment
(FPFi, n) and the sinking-fund factor with
equal-payment series (FFAi ,n)

A = P × FPA = P × FPF
× FFA
(B. 13)

A = P × (1 + i)n × i / (1 +
i)n - 1
(B. 14)

Figure B.4 Single Present Amount
and Equal-payment Series

A = P × i × (1 + i)n / (1 + i)n
- 1
(B. 15)

The resulting factor i x (1 +i)n / [(1
+i)n - 1] is known as the capital-recovery factor with
equal-payment series, and is written as (FFA', n). It is used to
calculate equal payments required to amortize a present amount of
a loan, where the interest is calculated on the balance. This
type of financial settlement is the basis of the majority of
loans and constitutes the most common form of amortizing a debt.

In many cases, the annual payments are not made
in an equal-payment series. In some countries it is common to
find geometric series of payments, that is, payments where each
term is equal to the previous one, multiplied by a factor:

(B. 16)

where S stands for the first payment and r, the
factor by which it is multiplied. This series can symbolize, for
example, a monthly indexed quota, with a monthly interest of a.
In this case, the initial quota is S. To be able to work with
this series through the known formulas, each of the quotas should
first be brought to present values:

(B. 17)

After removing the common S/r factor in the
series:

(B. 18)

the expression between parentheses is the sum of
the geometric ratio series a /(1 + i)

(B. 19)

Working with this expression it is possible to
obtain:

(B.20)

If instead of working with present value, the
annual quotas equivalent to the series are required, the
following equations should be used:

(B. 21)

In this way, whether using P or A, it is possible
to work easily with the known equations.

To simplify matters, the discussion of interest
has been based on interest periods of one year. However,
agreements may specify that interest be paid more frequently, for
example, biannually, quarterly, or monthly. Such agreements
result in interest periods of six months, three months or one
month, and the interest is compounded twice, four times or twelve
times for the year, respectively.

The interest rates associated with this method of
more frequent compounding are normally quoted on an annual basis
according to the following convention. When the effective rate of
interest is 4.8 % compounded every 6 months, the annual or
nominal interest is quoted as " 9.6 % annually, compounded
biannually". For a 2.4 % effective rate of interest
compounded at the end of each 3-month period, the nominal
interest is quoted as "9.6% annually, compounded quarterly".
Thus the nominal rate of interest is expressed on an annual basis
and this is determined by multiplying the effective rate of
interest by the number of interest periods in the year.

The effect of compounding more frequently is that
the effective rate of interest, is higher than the nominal rate
of interest. For example, consider a 9.6% nominal rate of
interest, compounded biannually. The value of a dollar at the end
of a year, when a dollar is compounded at 4.8% for each 6-month
period is:

F = US$ 1(1.048) (1.048) = US$ 1 (1.048)2
= US$ 1.0983

The real interest earned on the dollar for a
year, is equal to US$ 0.0983. As a result, the effective rate of
interest is 9.83 %. An expression for the annual effective rate
of interest can be derived from the last reasoning, that is:

i = nominal rate of interest (annual)

ieff = effective rate of interest
(per period)

c = number of interest periods per year

ieff = annual effective rate of
interest = (1 + i/c)c - 1
(B.22)

Example B.8 Effective Rate of Interest

Find the value of US$ 1 000 for 4 years at a 10%
nominal rate compounded quarterly.

As a limit, interest can be considered to be
compounded an infinite number of times per year, that is
continually. Under these conditions, continual annual effective
interest for interest compounded continuously is defined as:

ieff = lim (1 + i/c)c -
1
(B.23)

The right side of the equality is rearranged to
include i in the exponent:

(1 + i/c)c = (1 + i/c)i x c /i
(B. 24)

The value of the mathematical symbol e is the
value of (1 + l/n)n as n approaches infinity, then:

The behaviour of the world economy in the past
shows a general inflationary tendency in the cost of goods. This
tendency was reversed during specific periods, but from a global
point of view there seems to be an incessant pressure on prices
to increase. Low inflation rates seem to have little impact on
changes in prices, but when inflation exceeds 10% annually, it
can produce extremely serious consequences for individuals and
for institutions (see section 7.9).

Inflation is normally described in terms of an
annual or monthly percentage, which represents the rate at which
prices of goods in the year or month being considered increased
in relation to the prices of the previous year or month. Since
the rate is defined in this manner, inflation has a compounded
effect. Therefore, prices which inflate at a rate of 8% monthly
will increase by 8% in the first month and in the following month
the expected increase will be 8% of the new prices. Since the new
prices include the original 8% increase, the rate of increase is
applied to the 8% increase already added.

The same applies to the successive months and as
a consequence, the inflation rates are compounded in the same
manner as the interest rate. To incorporate the effects of
inflation in economic studies, interest factors must be used such
that inflationary effects can be identified in monetary values at
the different points in time. The standard procedure to avoid the
loss of buying power that accompanies inflation is:

Study all costs associated with a project
in terms of present monetary value.

Modify the costs estimated in step 1 so
that at each future date they represent the cost at that
time in terms of the monetary values that must then be
spent.

Calculate the equivalent quantity of cash
flow which results from step 2, considering the time
value of money (interest rate of the market).

It is important to observe that the interest rate
at which it is possible to invest in a financial or banking
operation represents the market interest rate (financial standard).
This interest rate is compounded by the inflation rate and the
opportunity of earning. If these two effects are separated; iR,
the rate which represents the money's earning power without
inflation, is related to i, the market rate and b , the inflation
rate, by Equation 7.25 of this manual as:

(1 + i) = (1 + ß) x (1 + iR)

iR = ((1 + i) / (1 + b )) - 1
(B. 28)

Example B.9 Real Rate of Interest

A person invests his money in a bank at an
interest rate of 25 % per year where the inflation rate is 20%
per year. What is the true or real interest rate?

Answer: From Equation B.28

1 + iR = (1+0.25)/(1+0.20) = 1.042

iR = 4.2%

This example shows that the effect of the
inflation is to make a business seems more profitable than it
actually is.